An opening stand of 47 by Rogers and Steve Stubbings gave the visitors a decent start before they lost the wickets of Stubbings, John Sadler and Dominic Telo in quick succession.
Rogers notched up his 50 from 74 balls just before lunch to steady the ship, and the revival continued after the restart.
But Glamorgan soon had another breakthrough, as Derbyshire went from 122-3 to 126-5, courtesy of two David Harrison wickets.
Dan Birch looked confident until he was caught out off Robert Croft, leaving his side 156-6.
Clare and Wagg added another 50 before Jamie Dalrymple dived full stretch to catch Wagg, then Jake Needham was out lbw, becoming Croft's 900th first-class victim for Glamorgan.
Langeveldt's 40 off just 19 balls at the close earned Derbyshire a second batting point, and Wagg's late wicket ensured a nervous end to the day for Glamorgan.